Bistability dynamics in some structured ecological models ∗

Jifa Jiang†

Department of Mathematics, Tongji University, Shanghai 200092, P.R.China

Email: jiangjf@mail.tongji.edu.cn

Junping Shi‡

Department of Mathematics, College of William and Mary, Williamsburg, VA 23185, USA

and, School of Mathematics, Harbin Normal University, Harbin, Heilongjiang 150080, P.R.China

Email: shij@math.wm.edu

Alternative stable states exist in many important ecosystems, and gradual change ofthe environment can lead to dramatic regime shift in these systems [BHC, M, RDRK, SCF].Examples have been observed in the desertification of Sahara region, shift in Caribbean coralreefs, and the shallow lake eutrophication [CLB, SC, SCF]. It is well-known that a social-economical system is sustainable if the life-support ecosystem is resilient [Ho2, FCW]. Hereresilience is a measure of the magnitude of disturbances that can be absorbed before a systemcentered at one locally stable equilibrium flips to another. Mathematical models have beenestablished to explain the phenomena of bistability and hysteresis, which provide qualitativeand quantitative information for ecosystem managements and policy making [CLB, PP].However most of these models of catastrophic shifts are non-spatial ones. A theory forspatially extensive, heterogeneous ecosystems is needed for sustainable management andrecovery strategies, which requires a good understanding of the relation between systemfeedback and spatial scales [FCW, WHCK, RDRK].

In this essay we survey some recent results on structured evolutionary dynamics includ-ing reaction-diffusion equations and systems, and discuss their applications to structuredecological models which display bistability and hysteresis. In Section 1, we review severalclassical non-spatial models with bistability; we discuss their counterpart reaction-diffusionmodels in Section 2, and especially diffusion-induced bistability and hysteresis. In Sec-tion 3, we introduce some abstract results and concrete examples of threshold manifolds(separatrix) in the bistable dynamics.

∗Keywords: bistability, multiple stable states, threshold, bifurcation, reaction-diffusion†Partially supported by Chinese NNSF grants 10671143 and 10531030‡Partially supported by United States NSF grants DMS-0314736, DMS-0703532, EF-0436318, Chinese

NNSF grant 10671049, and Longjiang scholar grant.

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1 Non-structured models

Logistic model was first proposed by Belgian mathematician Pierre Verhulst in 1838 [V]:

(1.1)dP

dt= aP

(1 − P

N

), a,N > 0.

Here a is the maximum growth rate per capita, and N is the carrying capacity. A moregeneral logistic growth type can be characterized by a declining growth rate per capitafunction. However it has been increasingly recognized by population ecologists that thegrowth rate per capita may achieve its peak at a positive density, which is called an Allee

effect (see Allee [Al], Dennis [De], Lewis and Kareiva [LK]). An Allee effect can be caused byshortage of mates (Hopf and Hopf [HH], Veit and Lewis [VL]), lack of effective pollination(Groom [Gr]), predator saturation (de Roos et. al. [dR]), and cooperative behaviors (Wilsonand Nisbet [WN]).

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0

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Figure 1: (a) logistic; (b) weak Allee effect; (c) strong Allee effect; the graphs on top row are growth

rate uf(u), and the ones on lower row are growth rate per capita f(u).

If the growth rate per capita is negative when the population is small, we call such agrowth pattern a strong Allee effect (see Fig.1-c); if f(u) is smaller than the maximum butstill positive for small u, we call it a weak Allee effect (see Fig.1-b). In Clark [Cl], a strongAllee effect is called a critical depensation and a weak Allee effect is called a noncritical

depensation. A population with a strong Allee effect is also called asocial by Philip [Ph].Most people regard the strong Allee effect as the Allee effect, but population ecologistshave started to realize that Allee effect may be weak or strong (see Wang and Kot [WK],Wang, Kot and Neubert [WKN]). The possible growth rate per capita functions were alsodiscussed in Conway [Co1, Co2]. A prototypical model with Allee effect is

(1.2)dP

dt= aP

(1 − P

N

)· P − M

|M | , a,N > 0.

2

If 0 < M < N , then the equation is of strong Allee effect type, and if −N < M < 0, thenit is of weak Allee effect type. At least in the strong Allee effect case, M is called sparsityconstant.

The dynamics of logistic equation is monostable with one globally asymptotically stableequilibrium, and the one of strong Allee effect is bistable with two stable equilibria. WeakAllee effect is also monostable, although the growth is slower at lower density. Anotherexample of weak Allee effect is the equation of higher order autocatalytic chemical reactionof Gray and Scott [GS]:

(1.3)da

dt= −kabp,

db

dt= kabp, k > 0, p ≥ 1.

Here a(t) and b(t) are the concentrations of the reactant A and the autocatalyst B, k isthe reaction rate, and p ≥ 1 is the order of the reaction with respect to the autocatalyticspecies. Notice that a(t) + b(t) ≡ a0 + b0 is invariant, then (1.3) can be reduced to

(1.4)db

dt= k(a0 + b0 − b)bp, k, a0 + b0 > 0, p ≥ 1,

which is of weak Allee effect type if p > 1, and of logistic type if p = 1. Autocatalyticchemical reaction has been suggested as a possible mechanism of various biological feedbackcontrol [Mu], and the similarity between chemical reaction and ecological interaction hasbeen observed since Lotka [L] in his pioneer work.

The cubic nonlinearity in (1.2) has also appeared in other biological models. Oneprominent example is the FitzHugh-Nagumo model of neural conduction [Fit, Na], whichsimplifies the classical Hodgkin-Huxley model:

(1.5) ǫdv

dt= v(v − a)(1 − v) − w,

dw

dt= cv − bw, ǫ, a, b, c > 0,

where v(t) is the excitability of the system (voltage), and w(t) is a recovery variable rep-resenting the force that tends to return the resting state. When c is zero and w = 0, (1.5)becomes (1.2). Another example is a model of the evolution of fecally-orally transmitteddiseases by Capasso and Maddalena [CM1, CM2]:

(1.6)dz1

dt= −a11z1 + a12z2,

dz2

dt= −a22z2 + g(z1), a11, a12, a22 > 0.

Here z1(t) denotes the (average) concentration of infectious agent in the environment; z2(t)denotes the infective human population; 1/a11 is the mean lifetime of the agent in theenvironment; 1/a22 is the mean infectious period of the human infectives; a12 is the multi-plicative factor of the infectious agent due to the human population; and g(z1) is the force ofinfection on the human population due to a concentration z1 of the infectious agent. If g(z1)is a monotone increasing concave function, then it is known that the system is monostablewith the global asymptotical limit being either an extinction steady state or a nontrivial

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endemic steady state. However if g(z1) is a monotone sigmoid function, i.e. a monotoneconvex-concave function with S-shape and saturating to a finite limit, then the system (1.6)possesses two nontrivial endemic steady states and the dynamics of (1.6) is bistable whichcan be easily seen from the phase plane analysis.

0

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0.6

0.8

1

V

0.1 0.2 0.3 0.4 0.5 0.6

r

Figure 2: Equilibrium bifurcation diagram of (1.8) with h = 0.1, where the horizontal axis is r and

the vertical axis is V .

Now we turn to some existing models which could lead to catastrophic shifts in ecosys-tems. In 1960-70s, theoretical predator-prey systems are proposed to demonstrate variousstability properties in systems of populations at two or more trophic levels [RM, Ro]. Asimplified model with predator-prey feature is that of grazing system of herbivore-plantinteraction as in Noy-Meir [No], see also May [M]. Here V (t) is the vegetation biomass, andits quantity changes following the differential equation:

(1.7)dV

dt= G(V ) − Hc(V ),

where G(V ) is the growth rate of vegetation in absence of grazing, H is the herbivorepopulation density, and c(V ) is the per capita consumption rate of vegetation by the her-bivore. If G(V ) is given by the familiar logistic equation, and c(V ) is the Holling type II(p = 1) or III (p > 1) functional response function [Ho1], then (1.7) has the form (afternondimensionalization):

(1.8)dV

dt= V (1 − V ) − rV p

hp + V p, h, r > 0, p ≥ 1.

This equation (with p = 2) also appears as the model of insect pests such as the sprucebudworm (Choristoneura fumiferana) in Canada and northern USA (see Ludwig et. al.[LJH]), in which V (t) is the budworm population. In either situation, the harvesting effortis assumed to be constant as the change of predator population occurs in a much slow time

scale compared to that of prey. The function c(V ) =γV p

hp + V pwith p ≥ 1 is called Hill

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function in some references. We notice that Hill function is one of sigmoid functions whichis defined in the epidemic model (1.6).

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V –0.6

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1

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V

Figure 3: (top) Graph of the growth rate function f(V ) = V (1 − V ) − rV p

hp + V pwith h = 0.1;

(bottom) Graph of the growth rate per capita f(V )/V . (a) r = 0.17 (left); (b) r = 0.2 (middle); (c)

r = 0.3 (right).

To describe the catastrophic regime shifts between alternative stable states in ecosys-tems, a minimal mathematical model

(1.9)dx

dt= a − bx +

rxp

hp + xp, a, b, r, h > 0,

is proposed in Carpenter et. al. [CLB], see also Scheffer et. al. [SCF]. (1.9) can be used inecosystems such as lakes, desert, or woodlands. For lakes, x(t) is the nutrients suspendedin phytoplankton causing turbidity, a is the nutrient loading, b is the nutrient removal rate,and r is the rate of internal nutrient recycling.

The equations (1.8) and (1.9) are examples of differential equation models which ex-hibit the existence of multiple stable states and the phenomenon of hysteresis. From thebifurcation diagrams (Fig. 2 for (1.8), and Fig. 4 for (1.9)), the system has three positiveequilibrium points when r ∈ (r1, r2) for some ∞ > r2 > r1 > 0, and the largest and smallestpositive equilibrium points are stable. For the grazing system (1.8), the number of stableequilibrium points changes with the herbivore density r. For low r, the vegetation biomasstends to a unique equilibrium slightly lower than 1 (the rescaled carrying capacity); as rincreases over r1, a second stable equilibrium appears through a supercritical saddle-nodebifurcation, and it represents a much lower vegetation biomass; as r continues to increasesto another parameter threshold r2 > r1, the larger stable equilibrium suddenly vanishesthrough a subcritical saddle-node bifurcation, and the lower stable equilibrium becomes theunique attracting one. As h increases gradually, the vegetation biomass first settles at ahigher level for low h, but it collapses to a lower lever as h passes r2; after this catastrophic

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1

2

3

4

5

6

7

8

x

0 1 2 3 4 5 6 7

r

Figure 4: Equilibrium bifurcation diagram of (1.9) with a = 0.5, b = 1, where the horizontal axis

is r and the vertical axis is x.

shift, even if h is restored slightly, the biomass remains at the low level unless h decreasesbeyond r1. This irreversibility of the hysteresis loop gives raise to a serious managementproblem for the grazing systems, see [No, M]. Similar discussions can be done for (1.9)as well as r decreases, see [SCF], where the drop from high density stable equilibrium tothe low one is called “forward shift”, and the recovery from the low one to high one is a“backward shift”.

It is worth pointing out that the S-shaped bifurcation curve in Fig. 2 and Fig. 4 canalso be viewed as a result of bifurcation with respect to conditions such as nutrient loading,exploitation or temperature rise [SCF]. That is a transition from a monostable system to abistable one, or mathematically, a cusp bifurcation from a monotone curve to a S-shapedone with two turning points (see Fig. 6). Such fold bifurcations have been discussed in muchmore general settings in Shi [Sh1], and Liu, Shi and Wang [LSW]. In general it is hard torigorously prove the exact transition from monostable to bistable dynamics, especially forhigher (including infinite) dimensional problems. In (1.8) with p = 2, one can show thecusp bifurcation occurs when h crosses h0 =

√3/27 ≈ 0.1925. A mathematical survey on

the fold and cusp type mappings (especially in infinite dimensional spaces) can be found inChurch and Timourian [CT].

We note that in Fig. 3-a and Fig. 5-c, the system is monostable with only one stableequilibrium point, yet the graph of “growth rate per capita”(see the lower graphs in Fig.3-a and Fig. 5-c) has two fluctuations before turning to negative. This is similar to theweak Allee effect defined earlier where the growth rate per capita changes the monotonicityonce. These geometric properties of the growth rate per capita functions motivate us toclassify all growth rate patterns according to the monotonicity of the function f(u)/u iff(u) is the gross growth rate in a model u′ = f(u):

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x

0

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2

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x

Figure 5: (top) Graph of the growth rate function g(x) = a − bx +rxp

hp + xpwith a = 0.5, b = 1;

(bottom) Graph of the growth rate per capita f(x)/x. (a) r = 2.5 (left); (b) r = 4 (middle); (c)

r = 5.5 (right).

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V

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r

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r

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1

V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

r

Figure 6: Cusp bifurcation in (1.8) with p = 2, where the horizontal axis is r and the vertical axis

is V . (a) h = 0.15 (left); (b) h =√

3/27 ≈ 0.1925 (middle); (c) h = 0.25 (right).

1. f(u) is of logistic type, if f(u)/u is strictly decreasing;

2. f(u) is of Allee effect type, if f(u)/u changes from increasing to decreasing when uincreases;

3. f(u) is of hysteresis type, if f(u)/u changes from decreasing to increasing then todecreasing again when u increases.

In all cases, we assume that f(u) is negative when u is large, thus f(u) has at least onezero u1 > 0. In the Allee effect case, if f(u) has another zero in (0, u1), then it is strongAllee effect, otherwise it is weak one; in the hysteresis case, if f(u) has another two zeros in(0, u1), then it is strong hysteresis, otherwise it is weak one. Here we exclude the degeneratecases when f(u0) = f ′(u0) = 0 (double zeros). Considering the ODE model u′ = f(u), theweak Allee effect or hysteresis dynamics appears to be no difference from the logistic casefor the asymptotic behavior, since f(u) > 0 for u ∈ (0, u1) and f(u) < 0 for u > u1. The

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definitions here are not only for mathematical interest. In the next section, we shall showthat the addition of diffusion to the equation can dramatically change the dynamics for theweak Allee effect or hysteresis.

2 Diffusion induced bistability and hysteresis

Dispersal of the state variable in a continuous space can be modeled by a partial differentialequation with diffusion (see Okubo and Levin [OL], Murray [Mu], Cantrell and Cosner[CC3]):

(2.1)∂u

∂t= d∆u + f(u), t > 0, x ∈ Ω.

Here u(x, t) is the density function of the state variable at spatial location x and time t,d > 0 is the diffusion coefficient, the habitat Ω is a bounded region in Rn for n ≥ 1,

∆u =

n∑

i=1

∂2u

∂x2i

is the Laplace operator, and f(u) represents the non-spatial growth pattern.

We assume that the habitat Ω is surrounded by a completely hostile environment, thus itsatisfies an absorbing boundary condition:

(2.2) u(x) = 0, x ∈ ∂Ω.

It is known (see [He]) that for equation (2.1) with boundary condition (2.2), there isa unique solution u(x, t) of the initial value problem with an initial condition u(x, 0) =u0(x) ≥ 0, provided that f(u), u0(x) are reasonably smooth. Moreover, if the solutionu(x, t) is bounded, then it tends to a steady state solution as t → ∞ if one of the followingconditions is satisfied: (i) f(u) is analytic; (ii) if all steady state solutions of (2.1) and(2.2) are non-degenerate (see for example, Polacik[Po] and references therein). Hence theasymptotical behavior of the reaction-diffusion equation can be reduced to a discussion ofthe structure of the set of steady state solutions and related dynamical behaviors. Thesteady state solutions of (2.1) and (2.2) satisfy a semilinear elliptic type partial differentialequation:

(2.3) d∆u(x) + f(u(x)) = 0, x ∈ Ω, u(x) = 0, x ∈ ∂Ω.

Since we are interested in the impact of diffusion to the extinction/persistence of popu-lation, we use the diffusion coefficient d as the bifurcation parameter. One can also use thesize of the domain Ω as an equivalent parameter. To be more precise, we use the change ofvariable y = x/

√d to convert the equation (2.3) to:

(2.4) ∆u(y) + f(u(y)) = 0, y ∈ Ωd, u(y) = 0, y ∈ ∂Ωd,

where Ωd = y :√

dy ∈ Ω. This point of view fits the classic concept of critical patchsize introduced by Skellam [Sk]. When Ω = (0, l), the one-dimensional region, the size of

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the domain is simply the length of the interval. In higher dimension, Ωd is a family ofdomains which same shape but “size” proportional to d−1/2. Here “size” can be definedas the one-dimensional scale of the domain. Size can also be defined through the principaleigenvalue of −∆ on the domain Ω with zero boundary condition, which is the smallestpositive number λ1(Ω) such that

(2.5) ∆φ(x) + λ1φ(x) = 0, x ∈ Ω, φ(x) = 0, x ∈ ∂Ω,

has a positive solution φ. Apparently λ1(Ωd) = λ1(Ω)/d. In application a habitat slowlyeroded by external influence can be approximated by such a family of domain Ωd withsimilar shape but shrinking size. This is a special case of habitat fragmentation. In thefollowing we use d as bifurcation parameter, and when d increases, the size (or the principaleigenvalue) of the domain Ωd decreases.

The multiplicity and global bifurcation of solutions of (2.3) have been considered bymany mathematicians in the last half century. Several survey papers and monographes canbe consulted, see for example [Am, CC3, Li, Sh2] and the references therein. In this sectionwe review some related results on that subject for the nonlinearity f(u) discussed in Section1 and their connection to ecosystem persistence/extinction.

For the Verhurst logistic model, the corresponding reaction-diffusion model was intro-duced by Fisher [Fis] and Kolmogoroff, Petrovsky, and Piscounoff [KPP] in 1937 in studyingthe propagation of an advantageous gene over a spatial region, and the traveling wave so-lution was considered. The boundary value problem

(2.6) d∆u + u(

1 − u

N

)= 0, x ∈ Ω, u = 0, x ∈ ∂Ω,

was studied by Skellam [Sk] when Ω = (0, L). Indeed in this case an explicit solution anddependence of L on D can be obtained via an elliptic integral [Sk]. When Ω is a generalbounded domain, it was shown (see Cohen and Laetsch [CL], Cantrell and Cosner [CC1],Shi and Shivaji [SS]) that there when 0 < d−1 < λ1(Ω) ≡ λ1, the only nonnegative solutionof (2.6) is u = 0, and it is globally asymptotically stable; when d−1 > λ1, (2.6) has a uniquepositive solution ud which is globally asymptotically stable. It is also known that ud(x) isis an decreasing function of d for d > λ−1

1 , and ud(x) → 0 as d−1 → λ+1 . Hence the critical

number λ1 represents the critical patch size. When the size of habitat gradually decreases,the biomass deceases too, and when it passes the critical patch size, the biomass becomeszero through a continuous change. Hence the bifurcation diagram of (2.6) is a continuousmonotone curve as shown in Fig.7 (a).

The bifurcation diagram in Fig.7 (a) changes when an Allee effect exists in the growthfunction f(u). For the boundary value problem

(2.7) d∆u + u(

1 − u

N

)· u − M

|M | = 0, x ∈ Ω, u = 0, x ∈ ∂Ω,

one can use M as a parameter of the bifurcation of the bifurcation diagrams. We alwaysassume M < N . When M ≤ −N , the growth rate per capita is decreasing as in logistic

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case, thus the bifurcation diagram is monotone as in Fig 7 (a). When −N < M < 0, thegrowth rate per capita is of weak Allee effect type, and a new type of bifurcation diagramappear (Fig 7 (b)). We notice that the nonlinearity in (2.7) is normalized so that the growthrate per capita at u = 0 is always 1 when M < 0. The rigorous mathematical results aboutexact multiplicity of steady state solutions and global bifurcation diagram Fig 7 (b) areobtained in Korman and Shi [KS], and Shi and Shivaji [SS] for more general nonlinearityand the domain being a ball in Rn. We also mentioned that if the dispersal does not satisfylinear diffusion law but a nonlinear one, then weak Allee effect can also occur, and thebifurcation diagram of steady state solutions is in like Fig. 7-b, see Cantrell and Cosner[CC2], and Lee et. al. [LSTS].

d−1

u

λ1(Ω)

d−1

u

λ1(Ω)λ∗(Ω)

d−1

u

λ∗(Ω)

Figure 7: Bifurcation diagrams for (2.7): (a) logistic (upper); (b) weak Allee effect (middle); (c)

strong Allee effect (lower).

Compared to the logistic case, a backward (subcritical) bifurcation occurs at (d−1, u) =(λ1, 0), and there is a new threshold parameter value 0 < λ∗ < λ1 exists. For d−1 < λ∗

(extinction regime), the population is destined to extinction no matter what the initial pop-ulation is; for d−1 > λ1 (unconditional persistence regime), the population always survive

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with a positive steady state. However in the intermediate conditional persistence regime,λ∗ < d−1 < λ1, there are exactly two positive steady state solutions u1,d and u2,d. In fact,it can be shown that the three steady state solutions (including 0) can be ordered so thatu1,d(x) > u2,d(x) > 0. Here u1,d and 0 are both locally stable, hence the diffusion effectinduces a bistability for a monostable model of weak Allee effect. A sudden collapse of thepopulation occurs if d increases (or the domain size decreases) when d−1 crosses λ∗, andthe system shift abruptly from u1,d to 0 and it is not recoverable. This may explain that insome ecosystem with weak Allee effect, a catastrophic shift could still occur although thecorresponding ODE model predicts unconditional persistence.

For 0 < M < N in (2.7), a strong Allee effect makes the bistability exists even forsmall diffusion case (d small). If N/2 ≤ M < N , u = 0 is the unique nonnegative solutionof (2.7) thus extinction is the only possibility. If 0 < M < N/2, there exist at least twopositive steady state solutions of (2.7) following a classical result of variational methodsby Rabinowitz [R1]. When the domain is a ball in Rn, it was shown by Ouyang and Shi[OS1, OS2] that (2.7) has at most two positive solutions and the bifurcation diagram isexactly like Fig.7-c. Earlier the exact bifurcation diagram for the one-dimensional problemwas obtained by Smoller and Wasserman [SmW]. It is well-known that in this case that asmall initial population always leads to extinction, thus a single threshold value λ∗ exists toseparate the extinction and conditional persistence regimes. Earlier work on the dynamicsof (2.1) and (2.2) with strong Allee effect was considered in Bradford and Philip [BP1, BP2]and Yoshizawa [Yo].

The exact multiplicity results proved in Ouyang and Shi [OS1, OS2] (see also Shi [Sh2])hold for more general nonlinearities f(u), and the criterion on f(u) for the exact multiplicityare given by the shape of the function f(u)/u and the convexity of f(u). Another exampleis the border line case for (2.7) between the weak (M < 0) and strong Allee effect (M > 0),or more generally, the equation of autocatalytic chemical reaction (1.4) (assuming thata0 + b0 = 1):

(2.8) d∆u + up(1 − u) = 0, x ∈ Ω, u = 0, x ∈ ∂Ω, p > 1.

The bifurcation diagram of (2.8) is similar to Figure 7-c, and a proof can be found in[OS1, OS2] or Zhao, Shi and Wang [ZSW]. Precise global bifurcation diagrams can alsobeen shown for the reaction-diffusion systems of autocatalytic chemical reaction (1.3) andepidemic model (1.6), and we will discuss them in the next section along with the associateddynamics.

The threshold value λ∗ is important biologically as λ∗ could give early warning ofextinction for the species. Usually it is difficult to give a precise estimate of λ∗ and it seemsthat there is no existing result on that problem. Here we only give an estimate of λ∗ forthe equation (2.7) with N = 1 and M ∈ (1/2, 1). Hence we consider

(2.9) d∆u + u(1 − u)(u − M) = 0, x ∈ Ω, u(x) = 0, x ∈ ∂Ω.

Here we have f(u) = u(1 − u)(u − M). From an idea in [SS], λ∗ > λ1/f∗, where f∗ =

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maxu∈[0,1] f(u)/u, or the maximal growth rate per capita. An upper bound of λ∗ canbe obtained if (2.9) has a nontrivial solution for that d. We define an associated energyfunctional

(2.10) I(u) =d

2

∫

Ω|∇u|2dx −

∫

ΩF (u)dx,

where F (u) =∫ u0 f(t)dt = −1

4u4 +

1 + b

3u3 − b

2u2. It is well-known that a solution u of

(2.9) is a critical point of the functional I(u) in certain function space (see Rabinowitz [R2]or Struwe [St] for more details.) In particular, if inf I(u) < 0, then (2.9) has a nontrivialpositive solution. For small d, it is apparent that inf I(u) < 0 if M ∈ (1/2, 1). Hence forlargest d = d so that inf I(u) < 0, we must have λ∗ < d−1. For the case Ω = (0, L), we canobtain that

(2.11)2π2

L2(1 + M)< λ∗ <

48

L2(3 − M).

Here the upper bound is obtained by using a test function u(x) = x/l for x ∈ [0, l], u(x) = 1for x ∈ [l, L/2] and u(x) = u(L − x) for x ∈ [L/2, L], then optimizing among all possiblevalue of l. The estimate (2.11) is indeed quite sharp. For example, for L = 1 and M = 0.2,the estimate (2.11) becomes 16.45 < λ∗ < 17.14. A numerical calculation using Maple andalgorithm in [LSTS] shows that λ∗ ≈ 16.61. The threshold value for other problems canbe estimated similarly, and in general the determination of the threshold value remains aninteresting open question.

Next we turn to bifurcation diagrams with hysteresis. The hysteresis diagrams inSection 1 (Fig. 2 and 4) are generated with parameter r, which is the herbivore density in(1.8) or the rate of internal nutrient recycling in (1.9). In this subsection, we consider thecorresponding reaction-diffusion models. First the steady state reaction-diffusion grazingmodel:

(2.12) d∆V + V (1 − V ) − rV p

hp + V p= 0, x ∈ Ω, V = 0, x ∈ ∂Ω,

was considered in Ludwig, Aronson and Weinberger [LAW]. For the case n = 1, by using thequadrature method, they show that the rough bifurcation diagram goes from a monotonecurve with a unique large steady state, to an S-shaped curve, to a disconnected S-shapedcurve, and finally a monotone curve with a unique small steady state, when r increases fromnear 0 to a large value (see Fig. 8 or the ones in [LAW]). Note that the bifurcation diagramsin [LAW] are not exact, and it is only shown that the equation has at least three positivesolutions but not exactly three. An exact multiplicity result like the one in [OS1, OS2] isnot known even when n = 1. But it is known that in Fig. 8-b, the upper bound of the lowerbranch is the first zero of f(u), and the lower bound of the upper branch is the smallest zeroof F (u) =

∫ u0 f(t)dt = 0 such that f(u) > 0; in Fig. 8-a, the lower turning point λ∗ → ∞ if

the positive local minimum value of f(u) tends to zero.

12

d−1

u

λ1(Ω)

d−1

u

λ1(Ω)

d−1

u

λ1(Ω)

Figure 8: Bifurcation diagrams for (2.12): (a) weak hysteresis, r small but close to the first break

point in ODE hysteresis loop, corresponding to f in Fig 3-a (upper); (b) strong hysteresis, corre-

sponding to f in Fig 3-b (middle); (c) “collapsed”, r larger than the second break point, correspond-

ing to f in Fig 3-c (lower).

The transition of rough bifurcation diagrams suggests a bistable structure exists forintermediate range of r (see Fig. 2) when the nonlinearity is of strong hysteresis type, buta bistable structure could also exist when r is smaller when the nonlinearity is of weakhysteresis type (see Fig. 8-a). Indeed the S-shaped bifurcation diagram implies a hysteresisloop even though the weak hysteresis nonlinearity is positive until the zero at the “carryingcapacity”. Hence this is a hysteresis induced by the diffusion. Back to the context ofshrinking habitat size, this suggests that for a seemingly safe ecosystem with the grazingis not too big so that the ODE model predicts a large stable equilibrium, the addition ofdiffusion can endanger the ecosystem if the habitat keeps shrinking, and a sudden drop tothe small steady state is possible if the habitat size passes a critical value. Note that wedo not exclude the possibility of catastrophic shift due to the increase of the grazing effectr, but the results in reaction-diffusion model offers another possible cause of such sudden

13

collapse to be the decreasing natural habitat of the vegetation.

For the model (1.9) of lake turbidity following [CLB, SCF], a reaction-diffusion modelcan also be proposed:

(2.13)

ut = d∆u + a − bu +rup

hp + up, t > 0, x ∈ Ω,

u(x, t) = 0, x ∈ ∂Ω,

u(x, t) = u0(x), t > 0, x ∈ Ω.

A similar argument can be made to offer another possible cause of the turbidity in shallowlakes: the shrinking has occurred for many freshwater lakes because of the expanding ofagriculture or industry. Here the bifurcation diagram of the steady state equation is notreadily available in the existing literature, but similar problems with S-shaped bifurcationdiagrams can be found in [BIS, DL, KL, W], to name a few. Indeed the nonlinearity f(u)in (2.13) is qualitatively similar to the one in (2.12) (comparing Fig. 3 and Fig. 5), hencetheir bifurcation diagrams are similar.

In all discussions so far, we have used Dirichlet boundary condition (u = 0 on theboundary). While diffusion plays an instrumental role in inducing bistability, the Dirichletboundary condition also plays an important role. In some rough sense, Dirichlet boundarycondition is much more “spatially heterogeneous” than Neumann boundary condition (orno flux, reflection boundary condition), and is more rigid than Neumann boundary condi-tion. Here we also comment briefly on reaction-diffusion models with Neumann boundarycondition:

(2.14)

∂u

∂t= d∆u + f(u), t > 0, x ∈ Ω,

∂u

∂n= 0, t > 0, x ∈ ∂Ω,

u(0, x) = u0(x) ≥ 0, x ∈ Ω.

A classical result of Matano [Ma], Casten and Holland [CH] is that (2.14) has no stablenonconstant equilibrium solution provided that the domain Ω is convex. A direct conse-quence is that the reaction-diffusion equation (2.14) has same number of stable equilibriumsolutions as the ODE u′ = f(u), hence diffusion does not induce “more”stability. Howeverthe geometry of the domain Ω is also an important factor in stability problem. Matano [Ma]shows that if f(u) is of bistable type, say f(u) = u(1−u2), then (2.14) has a stable noncon-stant equilibrium solution if Ω is dumbbell-shaped, see also Alikakos, Fusco and Kowalczyk[AFK] for more intricate results in that direction. Indeed it was recently shown that the ge-ometry of the domain is even important for the magnitude of the first non-zero eigenvalue ofLaplacian operator under Neumann boundary condition, see Ni and Wang [NW]. The workof Matano [Ma] is also extended to two species competition models (Matano and Mimura[MaM]) for nonconvex domains and cooperative models (Kishimoto and Weinberger [KW])for convex domains. More results on Neumann boundary value problems can be found inNi [Ni1, Ni2].

14

To summarize, we examine the reaction-diffusion ecological models of bistability orhysteresis in this section. When the diffusion coefficient d is small, or equivalently thehabitat is large, we show the existence of multiple spatial heterogeneous steady states, thusthe system possesses alternative stable spatial equilibrium solutions. Moreover, even whenthe non-spatial model is not bistable, the reaction-diffusion model may be bistable as weshow the bistable structure in the weak Allee effect or weak hysteresis case. Hence diffusionenhances the stability of certain states in such systems.

The bifurcation diagrams can also be explained with habitat size as the thresholdvalue. Indeed habitat fragmentation has been identified as one of possible causes of theregime shift in the ecosystems [WM]. Our results here provide theoretical evidence to thatclaim with the reaction-diffusion model approach. Other approaches on the implicationof spatial heterogeneity on the catastrophic regime shifts have been taken. van Nes andScheffer [VS] investigated the lattice models with same nonlinearities in (2.12) and (2.13),but their numerical bifurcation diagrams have r or a as bifurcation parameter, just same asthe ODE models (see Fig. 2 and Fig. 4). Bascompte and Sole [BS, SB] consider spatiallyexplicit metapopulation models to show the existence of extinction thresholds when a givenfraction of habitat is destroyed.

Another question is that when the existence of multiple steady states indicates bista-bility, the global dynamics of such system is still unclear. We present some mathematicalresults in that direction in the following section.

3 Threshold manifold

For the ordinary differential equation such as (1.2) with strong Allee effect, u = M isa threshold point so that the extinction and persistence depends on whether the initialvalue u0 < M or > M . Bistable dynamics in higher dimensional dynamics is separated bya separatrix or threshold manifold. Sometimes such dynamics is also called saddle pointbehavior [CM2, CW]. This can be illustrated by considering the classical Lotka-Volterracompetition model (in nondimensionalized form):

(3.1) u′ = u(1 − u − Av), v′ = v(B − Cu − v),

where A,B,C > 0 satisfy C > B > A−1 > 0. The system is bistable since it possesses twolocally stable equilibrium points (1, 0) and (0, B), and a separatrix—the stable manifold ofthe unstable coexistence equilibrium (u∗, v∗) = ((AB − 1)/(AC − 1), (C − B)/(AC − 1)),separates the basins of attraction of two stable equilibria, see Fig. 9. We also note that (3.1)possesses another invariant manifold connecting (1, 0), (0, B) and (u∗, v∗), called carryingsimplex, see more remarks about it in later part of this section.

An abstract mathematical result of threshold manifold is recently proved by Jiang,Liang and Zhao [JLZ]. They prove that in a strongly order preserving or strongly monotonesemiflow in a Banach space, if there are exactly two locally stable steady states, and any

15

Figure 9: Phase portrait of the competition model (3.1). The stable manifold of (u∗, v∗) (connecting

orbit from the origin) is the threshold manifold which separates the basins of attraction of two stable

equilibria; and the unstable manifold of (u∗, v∗) (connecting orbits from stable equilibria) is the

carrying simplex.

other possible steady state is unstable, then the set which separates the basins of attractionof two stable steady states is a codimension-one manifold (see more precise statement in[JLZ]). The scalar reaction-diffusion equation such as (2.1) and (2.2) generates a stronglymonotone semiflow in some function space, thus this result is immediately applicable tothe scalar reaction-diffusion equation. Hence the existence of a codimension-one manifoldfor Nagumo equation or all examples discussed in Section 2 with exactly two stable steadystate solutions follows from [JLZ]. The existence of the threshold manifolds rely on earlierresults of Takac [Ta1, Ta2]. We also mention that the earliest example of threshold manifoldwas given by McKean and Moll [McM], and Moll and Rosencrans [MR] when the Nagumoequation:

(3.2) ut = duxx + u(a − u)(u − b), x ∈ (0, L), u(0) = u(L) = 0,

where 0 < b < a, was considered. They also considered the case when the cubic functionis replaced by a piecewise linear function suggested by McKean [Mc], as an alternative ofFitzHugh-Nagumo model. We remark that the existence of exactly two stable steady statesolutions for (2.1) and (2.2) heavily depends on the geometry of the domain Ω. Most exactmultiplicity results in Section 2 hold for the ball domains but not general bounded domainΩ, as shown by Dancer [Da] in the example of dumbbell shaped domains. Similar remarkcan be applied to Neumann boundary value problem (2.14). For the convex domains Ω, the

16

bistable reaction-diffusion equation (2.14) with f(u) = u(1−u2) (Allen-Cahn equation frommaterial science) has exactly two stable steady state solutions u = ±1 from the results of[CH, Mat], hence the existence of a threshold manifold follows from [JLZ]. But for dumbbellshaped domain, it could have more stable steady state solutions from the result of Matano[Mat].

The two locally stable equilibrium points in Jiang-Liang-Zhao’s theorem can also bereplaced by one locally stable steady state and “infinity” which is locally stable. An ab-stract formulation of such kind has been obtained in Lazzo and Schmidt [LS], but concreteexamples have been shown much earlier. For a matrix population model, Schreiber [Sch]proves the existence of a threshold manifold that separates the initial values leading toextinction or unbounded growth. A more famous example in partial differential equationsis the Fujita equation (Fujita [Fu]):

(3.3) ut = d∆u + up, x ∈ Rn, p > 1.

Fujita [Fu] observed that for p > (n + 2)/(n − 2) and n ≥ 3, then the solution with certaininitial value blows up in finite time, while some other solution tends to zero as t → ∞.Since the solution of the ordinary differential equation u′ = up with p > 1 always blowsup, then the bistability in Fujita equation is a combined effect of diffusion (stabilize) andgrowth (blow up). Aronson and Weinberger [AW] obtained some criteria on the extinctionand blow-up of similar type equation, and they called the sensitivity of initial value betweenthe extinction and blow-up the “hair-trigger effect”. Mizoguchi [Mi] proves the existenceof the unique threshold between extinction and complete blow-up for radially symmetriccompactly-supported initial value, and the existence of a threshold manifold cannot directlyfollow from [LS] due to the lack of compactness when the domain is the whole space. Similarresults have also been proved for bounded domain, see for example Ni, Sacks, and Tavantzis[NST].

An intriguing question is whether such precise bistable structure is still valid for systemsof equations. When the system is still a monotone dynamical system, apparently this istrue. For example, it holds for the reaction-diffusion counterpart of (3.1): the diffusivecompetition system with two competitors and no-flux boundary condition:

(3.4)

ut = du∆u + u(1 − u − Av), t > 0, x ∈ Ω,

vt = dv∆v + v(B − Cu − v), t > 0, x ∈ Ω,∂u

∂n=

∂v

∂n= 0, t > 0, x ∈ ∂Ω,

u(0, x) = u0(x) ≥ 0, v(0, x) = v0(x) ≥ 0, x ∈ Ω.

Here du ≥ 0 and dv ≥ 0. While the steady states of (3.1) are still (constant) equilibriumsolutions of (3.4), it is known that any stable steady state of (3.4) is constant one if Ω isconvex from Kishimoto and Weinberger [KW]. Thus a threshold manifold of codimension-one exists when Ω is convex following [JLZ] although the dynamics on the threshold is notclear. In a more general setting, Smith and Thieme [ST] studied abstract two species (u, v)

17

competition systems with the origin being a repeller. Assume that the unique nontrivialboundary steady state on each axis is stable and there is a unique positive steady state, thenthey showed that there is an invariant threshold manifold through the positive steady stateseparating the attracting domains for both axis steady states. See Jiang and Liang [JL] andCastillo-Chavez, Huang and Li [CHL] for more about threshold manifold of bistability incompetition models. It should be noted that the results of [JLZ] are not valid for generalcompetition systems with more than two competitors.

However for non-monotone dynamical system, in general there is no such structure evenwith only two stable steady states. Some systems may inherit threshold structure from theirlimiting systems or subsystems. Consider the reaction and diffusion of the two reactants Aand B in an isothermal autocatalytic chemical reaction, we have the system:

(3.5)

at = DA∆a − abp, bt = DB∆b + abp, t > 0, x ∈ Ω,

a(x, t) = a0 > 0, and b(x, t) = 0, t > 0, x ∈ ∂Ω,

a(x, 0) = A0(x) ≥ 0, b(x, 0) = B0(x) ≥ 0, x ∈ Ω,

where a and b are the concentrations of the reactant A and the autocatalyst B, p > 1,DA and DB are the diffusion coefficients of A and B respectively, and Ω is a boundedreaction zone in Rn [GS]. It is known that when reactor Ω is a ball in Rn, (3.5) has eitheronly the trivial steady state (a0, 0), or exactly three non-negative steady state solutionswith two of them stable. Under the additional assumption of equal diffusion coefficients(DA = DB), Jiang and Shi [JS] shown that in the latter case, the global stable manifoldfor the intermediate steady state (a2, b2) is a codimension-one manifold which separates thebasin of attraction of the two stable steady states, and moreover every solution converges toone of three steady state solutions. Here we use the fact that the asymptotic limit of (3.5) isan autonomous scalar reaction-diffusion equation, which is a monotone dynamical system,see Chen and Polacik [CP], Mischaikow, Smith and Thieme [MST]. Although rather special,this is a rare example that the complete dynamics is known for a non-monotone dynamicalsystem in infinite dimensional space. A different bistability result for (3.5) in Rn is alsoobtained in Shi and Wang [SW] which uses some ideas form [AW].

Capasso and Wilson [CW] analyzed the spread of infectious diseases with a reaction-diffusion system:

(3.6)

u1t = d∆u1 − a11u1 + a12u2, t > 0, x ∈ Ω,

u2t = −a22u2 + g(u1), t > 0, x ∈ Ω,

u1(x, t) = u2(x, t) = 0, t > 0, x ∈ ∂Ω,

u1(x, 0) = U1(x) ≥ 0, u2(x, 0) = U2(x) ≥ 0, x ∈ Ω.

This system models random dispersal of the pollutant while ignoring the small mobility ofthe infective human population. Here u1(x, t) denotes the spatial density of the pollutant,and u2(x, t) denotes the density of the infective human population. With g(u) being the

18

monotone sigmoid function discussed in Section 1, the steady state equation can be reducedto

(3.7) d∆u1 − a11u1 +a12

a22g(u1) = 0, x ∈ Ω, u1 = 0, x ∈ ∂Ω.

The nonlinearity here f(u1) = −a11u1 + a12

a22g(u1) is of strong Allee effect using the term

introduced in the last subsection, hence under some reasonable conditions and Ω being aball, the bifurcation diagram of (3.7) is the one in Fig.7-c. This is shown in [CW] for thecase of n = 1, and the general case when n ≥ 2 can be deduced from the results in Ouyangand Shi [OS1]. Since (3.6) is a monotone dynamical system, then again (3.6) admits acodimension-one manifold which separates the basin of attraction of the two stable steadystates ([JLZ]), which confirms the conjecture in [CW]. But it is still not known that whetherevery solution on the threshold manifold converges to the intermediate steady state solution.

Even less is known about the dynamical behavior of FitzHugh-Nagumo system:

(3.8)

ǫvt = dv∆v + v(v − a)(1 − v) − w, t > 0, x ∈ Ω,

wt = dw∆w + cv − bw, t > 0, x ∈ Ω,

v(x, t) = w(x, t) = 0, t > 0, x ∈ ∂Ω,

v(x, 0) = V (x) ≥ 0, w(x, 0) = W (x) ≥ 0, x ∈ Ω.

Here dv > 0 and dw ≥ 0. When c = 0, it follows that w → 0, and the dynamics of (3.8) isreduced to that of Nagumo equation (3.2) (in higher dimensional domain). Since (3.2) hasthe saddle point behavior, then (3.8) still possesses this saddle point behavior for 0 < c ≪ 1by structural stability theory. For more general parameter ranges, the existence of multiplepositive steady state solutions of (3.8) is known, see for example Matsuzawa [Mat] for a nicesummary. Notice that (3.8) is not a monotone dynamical system, thus even the informationof stable steady state solutions cannot imply the saddle point behavior.

Threshold manifolds are a class of invariant manifolds in applied dynamical systems,and they are sensitively unstable in the dynamic sense as a small perturbation will shiftit to the basin of attraction of a stable equilibrium. If one reverses the time t to −t toa system with threshold manifold, then the manifold becomes an attracting manifold, orvice versa. For example, in the logistic model (1.1), if time is reversed, then it has theexactly same dynamical behavior as Fujita equation or the abstract formulation in Lazzoand Schmidt [LS]: both the origin and the infinity are stable and the carrying capacity Nbecomes a threshold point. Similarly, if one reverses the time in the classical Lotka-Volterracompetition system (3.1) without diffusion, then the origin and the infinity become stable,and there is a threshold manifold containing the boundary steady state (1, 0), (0, B) andcoexistence steady state on which “hair-trigger effect” occurs, which is deduced from Hirsch[Hir1] or an analysis for phase pictures. Of course it is not realistic to reverse the time inlogistic model or Lotka-Volterra competition system. Nevertheless, in logistic model (1.1)or Lotka-Volterra system (3.1), both the origin and the infinity are repellers, there is athreshold manifold separating the repelling domains for the origin and the infinity. Such a

19

threshold manifold plays the role of carrying capacity in the logistic model, so it is oftencalled Carrying Simplex.

The first example of carrying simplex was showed by Hirsch [Hir1] in his seminal paper.For a dissipative and strongly competitive Kolmogorov system:

(3.9)

dxi

dt= xiFi(x1, x2, ..., xn)

xi ≥ 0, i = 1, 2, ..., n,

Hirsch [Hir1] proved that if the origin is a repeller, then there exists a carrying simplexwhich attracts all nontrivial orbits for (3.9) and it is homeomorphic to probability simplexby radial projection. Note that the dissipation implies that the infinity is also a repeller.

Smith [Sm1] investigated C2 diffeomorphisms T on the nonnegative orthant K whichpossesses the properties (see the hypotheses in [Sm1]) of the Poincare map induced by C2

strong competition system

(3.10)

dxi

dt= xiFi(t; x1, x2, ..., xn)

xi ≥ 0, i = 1, 2, ..., n,

where Fi is 2π-periodic in t, Fi(t; 0) > 0, (3.10) has a globally attracting 2π-periodic so-lution on each positive coordinate axis. This implies that the origin is a repeller for Tand it has a global attractor Γ. He proved that the boundaries of the repulsion domainof the origin and the global attractor relative to the nonnegative orthant are compact un-ordered invariant set homeomorphic to the probability simplex by radical projection. Heconjectured both boundaries coincide, serving as a unique carrying simplex. Introducinga mild additional restriction on T , which is generically satisfied by the Poincare map ofthe competitive Kolmogorov system (3.10), Wang and Jiang [WJ] proved this conjectureand the unstable manifold of m−periodic point of T is contained in this carrying simplex.Diekmann, Wang and Yan [DWY] have showed the same result holds by dropping one of thehypotheses in Smith’s original conjecture such that the result is easier to use in competitivemapping. Hirsch [Hir3] introduces a new condition—strict sublinearity in a neighborhoodof the global attractor, to give a new existence criterion for the unique carrying simplex.The uniqueness of the carrying simplex is important in classifying the dynamics of lower di-mensional competitive systems, for example the 3-dimensional Lokta-Volterra competitionsystem (Zeeman [Z]). The classification of many three dimensional competitive mappings(see Davydova, Diekmann and van Gils [DDG, DDG1], Hirsch [Hir3] and references therein)are still open, and the uniqueness of the carrying simplex is one of the reasons.

We also remark that although if one reverse the time t to −t in n-dimensional compe-tition system (3.9), then the system becomes a monotone system with both the origin andthe infinity stable (under the assumption that the origin and the infinity are repellers). Butthis new system is not strongly monotone as required in [JLZ, LS], hence the existence of

20

the carrying simplex cannot follow from [JLZ, LS] except in the case of n = 2. Indeed thisis one of the main difficulties in [Hir1, WJ, DWY].

We conclude our discussion of threshold manifolds for a model of biochemical feedbackcontrol circuits. More details on the modeling can be found in, for example, Murray [Mu]or Smith [Sm2]. A segment of DNA is assumed to be translated to mRNA which in turn istranslated to produce an enzyme and it in turn is translated to another enzyme and so onuntil an end product molecule is produced. This end product acts on a nearby segment ofDNA to produce a feedback loop, controlling the translation of DNA to mRNA. Let x1 bethe cellular concentration of mRNA, let x2 be the concentration of the first enzyme, andso on, finally let xn be the concentration of their substrate. Then this biochemical controlcircuit is described by the system of equations

(3.11) x1′ = g(xn) − α1x1, xi

′ = xi−1 − αixi, 2 ≤ i ≤ n,

where αi > 0 and the feedback function g(u) is a bounded continuously differentiablefunction satisfying

0 < g(u) < M, g′(u) > 0, u > 0.

Hence it models a positive feedback. For the Griffith model [Gri] we have

(3.12) g(xn) =xp

n

1 + xpn

where p is a positive integer (the Hill coefficient). For the Tyson-Othmer model [TO] wehave

(3.13) g(xn) =1 + xp

n

K + xpn

where p is a positive integer and K > 1. The solution flow for (3.11) is strongly monotone(see [Sm2] for detail). The steady states for (3.11) are in one-to-one correspondence withsolutions of

(3.14) g(u) = αu

where α =∏

αi. Suppose that the line v = αu intersects the curve v = g(u) (u ≥ 0)transversally. Then every non-negative steady state for (3.11) is hyperbolic, which impliesthat the number of steady states for (3.11) is odd for either Griffith or Tyson-Othmer model.For most of biological parameters in Griffith or Tyson-Othmer model, there are exactlythree steady states (see Selgrade [Se1, Se2, Se3] and Jiang [J1, J2]). In this case, the leaststeady state and the greatest steady state are asymptotically stable and intermediate oneis a saddle point through which there is an invariant threshold manifold whose norm is

positive. In the multistable case, there are

[n − 1

2

]invariant threshold manifolds which

separate the attracting domains for stable steady states (see [JLZ]). From a general result

21

of Mallet-Paret and Smith [MS], we know that on each invariant threshold manifold everyorbit either converges to the saddle point or is asymptotic to a nontrivial unstable periodicorbit. For n ≤ 3, all orbits tend to the corresponding saddle point on threshold manifolds,which was proved by using topological arguments in Selgrade [Se1, Se2], Dulac criterionfor 3-dimensional cooperative system in Hirsch [Hir2] and Lyapunov function in Jiang [J2];for n ≥ 5, in the bistable case for Griffith or Tyson-Othmer model, there may exist Hopfbifurcation on the unique threshold manifold (see [Se3]). But for n = 4, whether there is anontrivial periodic orbit or not on threshold manifold is an open problem. In Jiang [J1], itwas proved that for 4-dimensional Griffith or Tyson-Othmer model all orbits are convergentto a steady state via Lyapunov method for parameters with biological significance.

Hetzer and Shen [HeS] added a third equation describing explicitly the evolution oftoxin, called an inhibitor, and obtained the equations from the classical Lotka-Volterraequations for two competing species (in rescaled form):

(3.15)

u = u(1 − u − d1v − d2w)

v = ρv(1 − fu − v)

w = v − (g1u + g2)w,

where d1, d2, ρ, f, g1, g2 > 0. Note that O(0, 0, 0), Ex(1, 0, 0), and Ey(0, 1, g−12 ) are non-

negative steady states of (3.15). Observing that O is a saddle, not a repeller. Hetzer andShen [HeS] studied the long-time behavior for (3.15) and the existence of threshold manifoldin bistable case, where they called a “thin separatrix” following [HSW, ST]. Jiang and Tang[JT] have given a complete classification for dynamical behavior for (3.15) and proved thatthe bistability occurs if and only if

(3.16) a∗ > 0, b∗ < 0, c∗ > 0, ∆∗ = (b∗)2−4a∗c∗ > 0, 2a∗+b∗ > 0, and a∗+b∗+c∗ > 0,

where a∗, b∗, c∗ are given by

a∗ = g1(1 − d1f), c∗ = g2(d1 +d2

g2− 1),

anda∗ + b∗ + c∗ = (1 − f)(d1g1 + d1g2 + d2).

In this case the system (3.15) has exactly two hyperbolic positive steady states, one isstable, denoted by E∗, the other is a saddle point, denoted by E∗. (3.15) has exactly twostable steady states Ey and E∗. The stable manifold for the saddle point E∗, which is a2-dimensional smooth surface, separates the basins of attraction for Ey and E∗, hence thissmooth surface is a threshold manifold.

The production of the various proteins in the biochemical control circuit model (3.11)is, of course, not instantaneous and it is reasonable to introduce time delays into theseterms. If one does so, (3.11) becomes a delay differential equation:

(3.17) x1′ = g(xn(t − rn)) − α1x1, xi

′ = xi−1(t − rj−1) − αixi, 2 ≤ i ≤ n,

22

with all delays ri positive. It is easy to see that all steady states for (3.17) are the sameas (3.11) and if a steady state for (3.11) is linearly stable (unstable) then it is also linearlystable (unstable) for (3.17) ([Sm2] p.111). Thus in the bistable case for (3.17), there isa codimension-one threshold manifold through a saddle point separating the attractingdomains for the two steady states. The only difference is that such a threshold manifold inthe space of continuous functions is infinite dimensional and less information is known forthe dynamics on the threshold manifold. The results are similar for the multistable case(see [JLZ]). Of course another way to have an infinite dimensional threshold manifold is toadd diffusion to a bistable (multistable) monotone ODEs or FDEs with no-flux boundarycondition on a smooth and convex domain, and codimension-one threshold manifold(s) stillexist(s) (see [JLZ]).

4 Concluding Remarks

Sharp regime shifts occur in some large-scale ecosystems such as lakes, coral reefs, grazedgrasslands and forests. Mathematical models have been set up to explain the sudden changesand hysteresis cycles in these systems. In this article, we review some of these models witha focus on the impact of spatial dispersal and habitat fragmentation. The rich dynamicsof these problems share some common mathematical features like multiple steady states,threshold manifold (separatrix), and non-monotone bifurcation diagrams. Mathematicaltools from partial differential equations, bifurcation theory, and monotone dynamical sys-tems have been applied and further developed in studying these important problems rootedfrom various applied areas.

Establishing the basic structure of multiple steady states and threshold manifold is thefirst step in a complete understanding of the bistable dynamics, regime shifts and ecosystemsresilience. The dynamics on the separatrix could be very complicated, and there are alsoevidence that bistability in a reaction-diffusion predator-prey system could imply existenceof more complex patterns (see [MPL1, MPL2, PML]). Another important question is how tomake early warning of the regime shifts. The bifurcation diagrams suggest that the regimeshifts occur at a saddle-node bifurcation points, at which the largest eigenvalue (principaleigenvalue) of the linearized system is zero. Near bifurcation points, the principal eigenvalueis small. It has been recognized that the principal eigenvalue at a steady state is related tothe return time, which is another definition of resilience of the system (see Pimm [Pi]). Thereturn time is how fast a variable that has been displaced from equilibrium returns to it. Forthe dynamical models described here, such return time to the equilibrium is characterizedby exp(λ1t), where λ1 is the principal eigenvalue at the equilibrium. Hence early warningfor regime shifts in large scale could be triggered by a change in return time, provided thatinformation on the return time is obtained from small scale experiments.

Acknowledgement: J.S. would like to thank Steve Cantrell, Chris Cosner and ShiguiRuan for invitation of giving a lecture in Workshop on Spatial Ecology: The Interplay

23

between Theory and Data, University of Miami, Jan. 2005, and this article is partiallybased on that lecture. Part of this work was done when the authors visit National TsinghuaUniversity in Dec. 2007, and they would like to thank Sze-Bi Hsu and Shin-Hwa Wang fortheir hospitality. The authors also thank the anonymous referee for many helpful commentsand suggestions which improve the earlier version of the manuscript.

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